(1.1) The theory of mirror symmetry emanated from string theory and has had a fundamental impact on algebraic geometry ever since the groundbreaking work of Candelas, de la Ossa, Green and Parkes [COGP91]. The mirror symmetry heuristic predicts that every complex Calabi-Yau manifold $X$ has a mirror partner $\check{X}$ of the same dimension whose complex geometry is equivalent, in a suitable sense, to the symplectic geometry of $X$, and vice versa. A celebrated application of these ideas was the prediction of the numbers of rational curves of fixed degree (more precisely, Gromov-Witten invariants) of the quintic threefold in [COGP91] by means of period integral calculations on the mirror partner. An important challenge in the theory of mirror symmetry is to give an exact definition of what it means to be a mirror pair of Calabi-Yau manifolds, and to devise techniques to construct such pairs.